A quick rant on the structure of math, and symbolic thought

I have been thinking about math, specifically about Godel’s theory, as read from Douglas Hofstadter’s “I am a strange loop”. I had a basic intuition, or a gist, of the actual meaning of the incompleteness theorem, but it was still outside my grasp. Perhaps it still is, but what I read from that book on the subject has given me major enlightenment, as it has instilled in me the notion that numbers are not just digits with patterns and interesting phenomena, or for many people “that stuff you use to balance a checkbook or count with”, but rather, can be used to model any system, because something special about numbers encodes all of the “Structure” that could possibly exist in any theorem, ever. This is truly mind-bending, and it gives me pause about how I looked at numbers.

I am starting to wonder if all of mathematics are simply ways to express “computation” on number systems or patterns of numbers – e.g. calculus, arithmetic, etc, all being different “ways” (perhaps crude, perhaps elegant, I’m not sure yet) to perform transformations on a structural system, that system being composed entirely of numbers. It’s as if numbers themselves are some encoding device for a meta-structure that exists in the fabric of reality. This is partially true, as the symbol you use doesn’t much matter, a 1 can be replaced by an “!”, 2 by “@”, etc…, so long as the operations remain intact. This is plainly obvious by looking at the unary number system, which is still intuitive to us (I, II, III, IV, V, etc…)

See for example:
I + II = III, I + IV = V, (IV + I) * II = X

…seems odd at first blush, yet it makes sense upon closer inspection – so then symbols aren’t all that meaningful, until we specifically project meaning on to them! This is certainly true with math: all algebraic symbols are meant to be mostly meaningless - as simple “stand-ins” for a concept, that don’t “get in the way”, and allow you to do the actual operations without the metadata of the symbol itself – which would be a distracting piece of information.

All of this is pretty hand-wavy, and feels a little like pseudo quantum mechanic type babble, but I think it’s an idea worth exploring, and it’s still more concrete than most jargony new-age talk. It’s likely this has been explored before, and even has mathematical models to explain it. I’d like to know more!

What really blows my mind though, is that we are creating new maths, that involve algebraic structures, much of which don’t rely entirely or much at all, on numbers, at least at first blush. They simply encode some abstract understanding, using basic symbols. I am thinking in particular of category theory. It is meant to bridge disparate maths together, and help understand a structural pattern to any math system. It calls into question what symbols are, what thought is, and even what numbers are. It all seems to be basic symbolic representation, but its become so abstract, it starts to bend back on itself, in that it becomes so hard to understand, because it’s like peeking into the brain while trying to use that very same brain to understand itself.Every system seems to be encoding some kind of “reflective identity” property that is triggered under certain “introspective” circumstances, and acts as a sort of blockade. This is kind of an analogue to the notion of reflection in computing, where objects can inspect and understand themselves (of course by a program written by a user, not the actual code itself, somehow becoming “magically aware”). Another tangent to this is the notion of a quine), which is a computer program that produces its own source code as output.

Anyway, enough babble mumbo jumbo - I’m glad to get that all out. I’m excited to learn more about the structural nature of numbers though, and how I can actually apply it to create something cool! I think understanding the ladder of abstraction will help make the application of these ideas easier.